Washington (CNN) - Immigrant rights advocates across the United States say they've seen few signs of the ICE raids that Trump administration officials had warned would begin Sunday.

But a senior administration official told CNN that parts of the ICE deportation operation began on Saturday and would be expanded into other cities over the coming days.

"I wouldn't expect a big splash that matches the hysteria we've heard over the last several days," that official said.

President Donald Trump insisted Monday that the raids had started and had been "very successful."

A senior immigration official who has seen the operation plans told CNN the list of target cities and individuals remains the same. The official had not received any details about total arrests so far. Officials previously have said ICE agents in 10 cities would be seeking 2,000 undocumented immigrants who'd been ordered removed from the United States.

Such ICE operations are not unprecedented. But it's uncommon for officials to talk about them so extensively before they occur. Some advocates questioned whether the President and his administration had released details about raids simply as a scare tactic. Others cautioned that it's too soon to say.

"It's very quiet. Let's hope it stays that way," said Jose Mario Cabrera of the Coalition for Humane Immigrant Rights of Los Angeles, speaking to CNN Sunday afternoon.

Jennaya Dunlap of the Inland Coalition for Immigrant Justice in Ontario, east of Los Angeles, also said she hadn't seen signs of sweeps.

"The way we see it with all the rumors and hysteria, we're telling the community that ICE is always conducting operations," she said Sunday. "This is nothing new. It's a daily reality for us. "

There also haven't been any confirmed reports of migrants being apprehended in Baltimore, Chicago or New York, immigrant advocacy groups and officials in those cities told CNN.

"We don't have any information that the ICE raids actually occurred," New York Gov. Andrew Cuomo told WNYC. "If he does have ICE raids, our law enforcement will not cooperate with them. I don't put it past him to actually deport people to make a political point, which would be reprehensible, but I wouldn't put it past him. But we see no evidence of it thus far."

Trump insists 'many were taken out on Sunday'

President Trump said Monday that planned immigration raids had begun over the weekend and resulted in the capture of many undocumented immigrants.

"People came into our country illegally," Trump said at the White House. "Many were felons. Many were convicted of crimes. Many, many were taken out on Sunday. You just didn't know about it."

Without providing any details about the number of arrests that occurred, Trump insisted that the raids were successful. He said he'd spoken to the head of Immigration and Customs Enforcement and that it was a "very successful day."

"But you didn't see a lot of it," Trump said. "I'm not sure they should be telling you, but it was a lot."

Officials have said raid are slated for Atlanta, Baltimore, Chicago, Denver, Houston, Los Angeles, Miami, New York and San Francisco. New Orleans is also on the list, but the city tweeted last week that US Immigration and Customs Enforcement said it would suspend operations through the weekend in areas hit by Tropical Storm Barry, which weakened to a tropical depression Sunday.

Acting US Citizenship and Immigration Services chief says he does not know details of ICE raids

Because these are targeted operations, there wouldn't be large numbers of agents visible, the senior immigration official said. The operations plan allowed apprehensions only between the hours of 6 a.m. and 10 p.m., the official said.

ICE officials told CNN the agency would not release specific details about operations "due to law-enforcement sensitivities and the safety and security of U.S. Immigrant and Customs Enforcement personnel."

"As always, ICE prioritizes the arrest and removal of unlawfully present aliens who pose a threat to national security, public safety and border security," the official said, while also noting that anyone who's violated immigration laws may be subject to arrest, detention and possible deportation.

ICE personnel have encountered one problem since the operation began, the senior administration official said: Potential targets of raids were not home when agents arrived. Those people may have relocated temporarily, the official said, as a result of the media coverage of the operation before it began -- including comments from President Donald Trump, who said the raids would start on Sunday.

Fear spreads in immigrant communities

Ken Cuccinelli, acting director of US Citizenship and Immigration Services, told CNN Monday that he didn't know how many people had been arrested in this operation because the acting director of ICE hadn't told him.

Asked why he didn't have details, Cuccinelli said, "presumably because operational details are kept contained within the agency executing the operation, as they should be."

On Sunday Cuccinelli characterized the raids as normal ICE business and pointed to statistics showing ICE has deported fewer people under President Trump than it did under President Barack Obama.

"This is their job every day. We've got compassionate, loyal ICE agents who are just doing their job," he said. "It shows you how far we've fallen in that it's become news that they would actually go deport people who have removal orders."

Word of the planned raids has sent fear rippling through immigrant communities.

Across the United States, advocacy groups have been canvassing neighborhoods, handing out fliers telling people what to do if ICE agents show up at their door.

Advocates told CNN they're not letting down their guard.

"I'm worried they may conduct sweeps Monday or Tuesday," Cabrera said. "They may not. But lying to us could be part of the game."

New York's Legal Aid Society said they were thankful the ICE raids so far had been unsuccessful.

"However, we remain on high alert as we know that this administration will stop at nothing to carry out their xenophobic deportation agenda," the society said in a statement Monday.

Alejandra Lopez of the Legal Aid Society told CNN's Polo Sandoval that calls have been streaming in to the society's immigration helpline.

The majority of them, she said, came from frightened children who are US citizens and were trying to help their immigrant parents prepare for the raids.